Theorem zarmxt1 33879

Description: The Zariski topology restricted to maximal ideals is T₁ . (Contributed by Thierry Arnoux, 16-Jun-2024.)

Hypotheses

Ref	Expression
zartop.1	⊢ 𝑆 = (Spec‘𝑅)
zartop.2	⊢ 𝐽 = (TopOpen‘𝑆)
zarmxt1.1	⊢ 𝑀 = (MaxIdeal‘𝑅)
zarmxt1.2	⊢ 𝑇 = (𝐽 ↾t 𝑀)

Assertion

Ref	Expression
zarmxt1	⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Proof of Theorem zarmxt1

Dummy variables 𝑖 𝑗 𝑘 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zartop.1	. . . 4 ⊢ 𝑆 = (Spec‘𝑅)
2		zartop.2	. . . 4 ⊢ 𝐽 = (TopOpen‘𝑆)
3	1, 2	zartop 33875	. . 3 ⊢ (𝑅 ∈ CRing → 𝐽 ∈ Top)
4		zarmxt1.1	. . . 4 ⊢ 𝑀 = (MaxIdeal‘𝑅)
5	4	fvexi 6920	. . 3 ⊢ 𝑀 ∈ V
6		zarmxt1.2	. . . 4 ⊢ 𝑇 = (𝐽 ↾t 𝑀)
7		resttop 23168	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → (𝐽 ↾t 𝑀) ∈ Top)
8	6, 7	eqeltrid 2845	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ V) → 𝑇 ∈ Top)
9	3, 5, 8	sylancl 586	. 2 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Top)
10		eqid 2737	. . . . . . . . . . 11 ⊢ (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
11	10	mxidlprm 33498	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
12	11	ex 412	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝑚 ∈ (MaxIdeal‘𝑅) → 𝑚 ∈ (PrmIdeal‘𝑅)))
13	12	ssrdv 3989	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (MaxIdeal‘𝑅) ⊆ (PrmIdeal‘𝑅))
15		eqid 2737	. . . . . . 7 ⊢ (PrmIdeal‘𝑅) = (PrmIdeal‘𝑅)
16	14, 4, 15	3sstr4g 4037	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ (PrmIdeal‘𝑅))
17		sseq2 4010	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑙 → (𝑖 ⊆ 𝑗 ↔ 𝑖 ⊆ 𝑙))
18	17	cbvrabv 3447	. . . . . . . . . . . 12 ⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙}
19		sseq1 4009	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑘 → (𝑖 ⊆ 𝑙 ↔ 𝑘 ⊆ 𝑙))
20	19	rabbidv 3444	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑙} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
21	18, 20	eqtrid 2789	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
22	21	cbvmptv 5255	. . . . . . . . . 10 ⊢ (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (𝑘 ∈ (LIdeal‘𝑅) ↦ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙})
23	1, 2, 15, 22	zartopn 33874	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
24	23	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) ∧ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽)))
25	24	simpld 494	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)))
26		toponuni 22920	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘(PrmIdeal‘𝑅)) → (PrmIdeal‘𝑅) = ∪ 𝐽)
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → (PrmIdeal‘𝑅) = ∪ 𝐽)
28	16, 27	sseqtrd 4020	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 ⊆ ∪ 𝐽)
29		simpl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ CRing)
30	29	crngringd 20243	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑅 ∈ Ring)
31		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ 𝑇)
32	6	unieqi 4919	. . . . . . . . . . . 12 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾t 𝑀)
33	31, 32	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ ∪ (𝐽 ↾t 𝑀))
34	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝐽 ∈ Top)
35		eqid 2737	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	restuni 23170	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
37	34, 28, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑀 = ∪ (𝐽 ↾t 𝑀))
38	33, 37	eleqtrrd 2844	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ 𝑀)
39	38, 4	eleqtrdi 2851	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (MaxIdeal‘𝑅))
40		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
41	40	mxidlidl 33491	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (LIdeal‘𝑅))
42	30, 39, 41	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ (LIdeal‘𝑅))
43		eqid 2737	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
44	22, 43	zarclssn 33872	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) → ({𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ↔ 𝑚 ∈ (MaxIdeal‘𝑅)))
45	44	biimpar 477	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑚 ∈ (LIdeal‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
46	29, 42, 39, 45	syl21anc 838	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} = ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚))
47	22	funmpt2 6605	. . . . . . . 8 ⊢ Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})
48		fvex 6919	. . . . . . . . . . 11 ⊢ (PrmIdeal‘𝑅) ∈ V
49	48	rabex 5339	. . . . . . . . . 10 ⊢ {𝑙 ∈ (PrmIdeal‘𝑅) ∣ 𝑘 ⊆ 𝑙} ∈ V
50	49, 22	dmmpti 6712	. . . . . . . . 9 ⊢ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (LIdeal‘𝑅)
51	42, 50	eleqtrrdi 2852	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
52		fvelrn 7096	. . . . . . . 8 ⊢ ((Fun (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) ∧ 𝑚 ∈ dom (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
53	47, 51, 52	sylancr 587	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ((𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})‘𝑚) ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
54	46, 53	eqeltrd 2841	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}))
55	24	simprd 495	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → ran (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) = (Clsd‘𝐽))
56	54, 55	eleqtrd 2843	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝐽))
57	38	snssd 4809	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ⊆ 𝑀)
58	35	restcldi 23181	. . . . 5 ⊢ ((𝑀 ⊆ ∪ 𝐽 ∧ {𝑚} ∈ (Clsd‘𝐽) ∧ {𝑚} ⊆ 𝑀) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
59	28, 56, 57, 58	syl3anc 1373	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘(𝐽 ↾t 𝑀)))
60	6	fveq2i 6909	. . . 4 ⊢ (Clsd‘𝑇) = (Clsd‘(𝐽 ↾t 𝑀))
61	59, 60	eleqtrrdi 2852	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ ∪ 𝑇) → {𝑚} ∈ (Clsd‘𝑇))
62	61	ralrimiva 3146	. 2 ⊢ (𝑅 ∈ CRing → ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇))
63		eqid 2737	. . 3 ⊢ ∪ 𝑇 = ∪ 𝑇
64	63	ist1 23329	. 2 ⊢ (𝑇 ∈ Fre ↔ (𝑇 ∈ Top ∧ ∀𝑚 ∈ ∪ 𝑇{𝑚} ∈ (Clsd‘𝑇)))
65	9, 62, 64	sylanbrc 583	1 ⊢ (𝑅 ∈ CRing → 𝑇 ∈ Fre)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436  Vcvv 3480   ⊆ wss 3951  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  dom cdm 5685  ran crn 5686  Fun wfun 6555  ‘cfv 6561  (class class class)co 7431  Basecbs 17247   ↾_t crest 17465  TopOpenctopn 17466  LSSumclsm 19652  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231  LIdealclidl 21216  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  Frect1 23315  PrmIdealcprmidl 33463  MaxIdealcmxidl 33487  Speccrspec 33861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-dju 9941  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-rest 17467  df-topn 17468  df-0g 17486  df-topgen 17488  df-mre 17629  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-lpidl 21332  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-t1 23322  df-prmidl 33464  df-mxidl 33488  df-idlsrg 33529  df-rspec 33862

This theorem is referenced by: (None)